With the definition of open subsets of \(\mathbb {R}^{m|k}\)  at hand, one can define a skeletal supermanifold \(\mathcal {M}\)  of dimension \(m|k\)  as follows: it is an ordinary manifold \(|\mathcal {M}|\)  of dimension \(m\)  endowed with a sheaf of super algebras \(\mathcal {O}_\mathcal {M}\)  (the sheaf of functions) over \(|\mathcal {M}|\)  such that, for any open subset \(|U|\)  of \(|\mathcal {M}|\)  with a chart \(|U|\longrightarrow |\Omega |\subset \mathbb {R}^m\) , the superalgebra \(\Gamma (|U|,\mathcal {O}_{|U|})\)  of sections of \(\mathcal {O}_\mathcal {M}\)  over \(|U|\)  is isomorphic to some \(\mathcal {C}^\infty (\Omega ) = \mathcal {C}^\infty (|\Omega |)[\theta ^1,\cdots ,\theta ^k]\) , where \(\Omega \)  is an open subset of \(\mathbb {R}^{m|k}\) . Note that the algebra \(\mathcal {C}^\infty (|U|)\simeq \mathcal {C}^\infty (|\Omega |)\)  of smooth functions on the ordinary open subset \(|U|\)  coincides with \(\mathcal {C}^\infty (\Omega )/ \mathcal {J}_\Omega \) , where \(\mathcal {J}_\Omega \)  is the ideal of nilpotent elements of \(\mathcal {C}^\infty (\Omega )\) . This allows to recover the sheaf of smooth functions \(\mathcal {O}_{|\mathcal {M}|}\)  from \(\mathcal {O}_{\mathcal {M}}\)  and hence the manifold \(|\mathcal {M}|\) . We shall not developp this description further, since most of the examples studied in this text requires only to understand \(\mathbb {R}^{m|k}\)  (see [1]} for details).
